All Questions
2,027 questions with no upvoted or accepted answers
1
vote
0
answers
514
views
Minimizing the Frobenius norm of a quadratic matrix expression
Given matrices $R \in \mathbb R^{m \times n}$ and $Y \in \mathbb R^{p \times n}$, where $R$ is full rank, how can I solve the following optimization problems?
$$\min_{X \in \mathbb R^{p \times m}} \| ...
- 11
1
vote
0
answers
121
views
Algorithm for the nilpotence of matrix polynomials
Let $P$ be a multivariate polynomial of real-valued $N \times N$ matrices. Given $X_1, X_2, ..., X_M \in \mathcal{M}_N\{\mathbb{R}\}$, is there any optimal algorithm to determine whether the result of ...
- 19
1
vote
0
answers
151
views
Parameter of Brauer algebra
Let $O(V)$= the set of orthogonal transformation from the vector space $V$ to $V$ where $\dim V=n$. We know that the centralizer algebra of $O(V)$ on the tensor space $V^{\otimes{f}}$ is Brauer ...
- 179
1
vote
0
answers
107
views
Reference request concerning splitting fields for groups that are related to special symmetric groups
Denote the symmetric group of order $n!$ by $S_n$. Let $H:=S_p$ for an odd prime $p$.
Every finite field $k$ is a splitting field $(^*)$ for $kH$, in particular $k:=\mathbb{F}_p$.
Questions:
Is $k:=\...
- 311
1
vote
0
answers
21
views
Weakening s-unitality
Recall that a (non-unital, non-commutative) ring $R$ is left s-unital if for every $r\in R$, we have $r\in Rr$.
Consider the following conditions:
There is a nonzero integer $m$ such that for all $r\...
- 1,338
1
vote
0
answers
99
views
Extension of closed-form solvability of polynomial equations of x and exp to rational expressions?
Is my conjecture below true?
It's a new conjecture. It's an extension of Lin's theorem in [Lin 1983] which is cited in [Chow 1999] - see both references at the bottom of this page.
It seems that Ferng-...
- 1,053
1
vote
0
answers
163
views
When every localization of the polynomial ring over a ring has finitely many idempotents
Let $R$ be a commutative ring such that every localization ring $R_r$ has finitely many idempotents for each non nilpotent element $r\in R$. Why dose every localization ring $R[x]_{f(x)}$ have ...
- 11
1
vote
0
answers
106
views
Field theory, Abel-Ruffini theorem, technical question
Let me put the question first.
Let $F,K$ be subfields of $\mathbb{C}$. Suppose that $t,\rho\in \mathbb{C}$ are algebraic over $F$ and $\rho \in K$. If $F(t)\cap K\subset F$, is it true that $F(t,\rho)\...
- 11
1
vote
0
answers
30
views
Solve linear overdetermined system from subsystems that compose it
This is my first MathOverflow post: I apologize if my message is lacking of something. I also posted this question in Mathematics Stack Exchange, but as I haven't seen an answer I post it here.
...
1
vote
0
answers
35
views
Can an orderless set of inner product between N vectors determine unique structure of the vectors?
Suppose we have n vectors {a1,a2,a3,...,an} such that the sum of them is zero vector
a1+a2+a3+...+an=0
Now, we compute the inner product of each two vectors of them, i.e. we compute the Gramian matrix ...
- 31
1
vote
0
answers
164
views
Symmetric functions for multidimensional variables
I have $N$ variables (let's call them $X$) of dimensionality $D$, that I want to make permutation invariant.
For $D=1$ I know I can use e.g. the elementary symmetric polynomials to accomplish this. ...
- 11
1
vote
0
answers
167
views
When localization is indecomposable
We know that if $R $ is a domain then any localization of $R $ at any multiplicative subset of $R $ is indecomposable, that is, has no non trivial idempotents. Now let $R $ be a commutative ring with ...
- 29
1
vote
0
answers
188
views
Phase angles of a complex eigenvector
I have the following system for $\lambda \in \Bbb C, \lambda \neq 0$ and $\pmb{p},\pmb{q} \in \Bbb C^n$, $(\pmb{p}^T, \pmb{q}^T)\neq0$:
$$\begin{cases} F(\lambda) \pmb{p} - g(\lambda) \pmb{q} - \...
- 111
1
vote
0
answers
443
views
Eigenvalues of symmetric tridiagonal matrices with identical off diagonal elements
Is there a simple analytical solution to obtain eigenvalues (and eigenvectors) for this type of tridiagonal matrices ? ( Off diagonal elements are identical and the matrix is symmetric)
$$
\begin{...
- 111
1
vote
0
answers
59
views
Diagonally similar to submatrix of orthogonal matrix
Given $A \in \mathbb{R}^{n \times n}$, with $0 < |\det(A)| < 1$. Does a diagonal matrix $D$ exist such that
$$
B = D^{-1} A D
$$
is the principal submatrix of an $(n+1)\times(n+1)$ orthogonal ...
- 909
1
vote
0
answers
126
views
Fastest way to calculate the eigenvalues of a product of two Toeplitz matrices
I have the following problem:
I need to find the fastest way to calculate the eigenvalues of a matrix that is the product of two Toeplitz matrices. $B = A U$.
The first is a regular Toeplitz matrix $A$...
- 11
1
vote
0
answers
77
views
Compatibility with multiplication of a cyclic order on a ring
I am copying my question from here: https://math.stackexchange.com/q/3233462/427611.
Is it correct that $\mathbb Z/3\mathbb Z$ and $\mathbb Z/4\mathbb Z$ are the only rings with three or more ...
- 133
1
vote
0
answers
32
views
Probability of marking at least one row in given matrix
Let there be a matrix $\alpha=(a_{i,j})_{i\in [m], j\in [n]}$, where $a_{i,j}\in\{0,1\}$ And every row has exactly $r\le n$ ones.
We independently with probability $p$ choose some columns from this ...
- 123
1
vote
0
answers
60
views
Universal bimodule for homotopy biderivations
Recall the commutative story: for a commutative algebra $A$, its module of differentials $\Omega (A)$ is characterized by the universal property that any derivation $\delta \colon A \to M$ is in a ...
- 765
1
vote
0
answers
210
views
Algebraic relation given by a 3x3 determinant
I just encountered a very curious relation in an algebra. A bit simplified, I am working in a (particular) non-commutative algebra, with some relations.
One particular relation is the following:
For (...
- 15.8k
1
vote
0
answers
70
views
Minimum rank of a product of two block diagonal matrices with an arbitrary matrix
Let us assume that we have an arbitrary full-rank $l\cdot b \times l\cdot p$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), an $m \times ...
- 61
1
vote
0
answers
175
views
Abelian subgroup of maximal order
Let $\mathbf{F}_q$ be a finite field of order $q$ ($q$, an odd prime, or a power of the same). I know that for the matrix algebra $M(n,\mathbf{F}_q)$ (or $gl(n,q)$), the maximal dimension for a ...
- 175
1
vote
0
answers
72
views
Simplicial differential graded algebra and a filtration
Let $A$ be a simplicial differential algebra, i.e. for each $n \in \mathbb{N}$ a differential graded algebra $(A_n,d_n)$ and for each weakly increasing map $f \colon [n] \to [m]$ a morphism $f_* \...
- 154
1
vote
0
answers
164
views
When a finite codimensional subalgebra contains a finite codimension ideal?
What is a classification of all algebras $A$ (purely algebraic algebras, Banach or $C^*$ algebras or Lie algebras) with the following property:
Every finite codimensional subalgebra $B$ of $A$ ...
- 356
1
vote
0
answers
116
views
Full-rank Hadamard product given a certain structure
Let us assume that we have a full-rank randomly chosen $k\times (m\cdot l)$ matrix, $\boldsymbol{H}$, with
$l \leq k \leq (m\cdot l)$ and no specific structure (e.g., a realization of an IID complex ...
- 61
1
vote
0
answers
74
views
Property of a semisimple Lie algebra over complex number field
Is the following property true？
Let $L$ be a finite-dimensional Lie algebra over $\mathbb{C}$. Then $L$ is semisimple if and only if for every $x \in L$, there exists $y, z\in L$, such that $x=[y, z]...
- 165
1
vote
0
answers
98
views
Construct $A_\infty$ bimodules maps from dg-maps
Let $ A $ be a dg-algebra. Let $U,V,W$ and $Z$ be dg-bimodules over $A$-$A$. Suppose I have cofibrant replacements $\pi_U : Up \rightarrow U$ (as right dg-module) and $\pi_W : pW \rightarrow W$ (as ...
- 71
1
vote
0
answers
90
views
$S$ subring of $R$ implies $\mathrm{rank}_R M\leq \mathrm{rank}_S M$?
This is a question that I've posted in MSE but I've got no answers.
The statement is as follows:
Let $R$ be a ring with identity, and $S$ be a subring with $1_S=1_R$. Let $M$ be a free unitary $R$-...
1
vote
0
answers
105
views
formal smoothness and McQuillan formal schemes
Let $k$ be an algebraically closed field, $A\rightarrow B$ be a continuous map of weakly admissible topological $k$-local algebras.
We assume that it is formally smooth and topologically of finite ...
- 3,472
1
vote
0
answers
150
views
Minimax optimization of diagonal entries of function of matrix
Let $\mathbf{A}$ and $\mathbf{U}$ be arbitrary complex $M\times N$ and $N\times M$ matrices, respectively. Let denote superscript $(\cdot)^{\dagger}$ and $(\cdot)^{\mathrm{H}}$ as pseudo-inverse and ...
- 287
1
vote
0
answers
448
views
Smallest eigenvalue for large kernel matrix
I am interested in the the asymptotics of the minimum eigenvalue $\lambda_n^n$ of a class of kernel matrix $P = [ K(x_i - x_j) ]_{i,j}$, with $x_i$ equally spaced in the unit cube of $\mathbb{R}^d$.
...
- 151
1
vote
0
answers
52
views
About means of positive definite matrices
This question is related to this recent one.
Recall that on the cone $SPD_n$ of $n\times n$ symmetric positive definite matrices, there are various notions of means, which extend the same notions ...
- 52.3k
1
vote
0
answers
179
views
Matrix factorizations over $GL_2$ of a real quadratic ring of integers
tl;dr: The groups $GL_2(K)$, or $SL_2(K)$, where $K = \mathbb{C,R}$ admits several factorizations (the polar decomposition,
the KAN decomposition, the Schur triangular form, etc). Those
...
- 1,090
1
vote
0
answers
422
views
Integral of matrix determinant with respect to Lebesgue measure
$\newcommand\norm[1]{\lVert#1\rVert}
\newcommand\opnorm[1]{\norm{#1}_{\text{op}}}
\newcommand\Frnorm[1]{\norm{#1}_{\text F}}$Define
\begin{align*}
S_t=\{
(A,B)\in\mathbb{R}^{n\times n}\times\...
- 1,495
1
vote
0
answers
49
views
Permutation of eigenvalues induced by a loop
A friend of mine just mention me what I think is a very fun phenomena and I would be very interested to learn more about it:
Let $A,B\in \mathbb{C}^{n\times n}$ two matrices. And let $\lambda_1(z), \...
- 4,361
1
vote
0
answers
132
views
Transformations preserving the number of distinct eigenvalues
Let $A\in\mathbb{R}^{n\times n}$ be an $n\times n$ symmetric, invertible matrix with nonnegative real entries, $\mathbf{1}$ be the all one $n$-dimensional vector, and $\mathrm{diag}(v)$, $v=[v_1,v_2,\...
- 2,712
1
vote
0
answers
152
views
solving a non-linear Matrix equation
I am working on a problem of optimization of wireless sensor networks in order to achieve the optimal power allocation in the network. I find an equality matrix problem that can be written as ...
- 377
1
vote
0
answers
156
views
Completion of a ring and the canonical map
Given a noetherian ring $R$ and an ideal $I$.
Questions: Can $R$ be isomorphic to the $I$-adic complection $\hat{R}$ of $R$ without the canonical map $R \rightarrow \hat{R}$ being an isomorphism?
A ...
- 26.5k
1
vote
0
answers
94
views
Application of cup-product for Leibniz cohomology
In 1995, Loday introduced a cup-product operation on the graded cohomology of Leibniz algebra and showed that the cup-product operation satisfies the graded Zinbiel relation.
My question is how this ...
- 596
1
vote
0
answers
101
views
Controlling the rank of a Matrix product
Let $\bf{Q}$ be an $(m-k)\times m$ matrix satisfying $\bf{QQ}^H=\bf{I}$, $\bf{W}$ an $m\times m\ell$ matrix of the form
$$\bf{}W=\left[\begin{array}{ccccc}{\bf{w}}_1 &\bf{0} &\bf{0}&\...
1
vote
0
answers
49
views
Weakly reflexive algebra vs proper (residually finite-dimensional) algebra
Currently I am reading the book "Hopf Algebras. An Introduction" by S. Dascalescu, C. Nastasescu, S. Raianu. There is a Definition 1.5.20 on page 44 (boldface is mine):
An algebra $A$ is called ...
- 519
1
vote
0
answers
64
views
Conjugacy classes and normal form of $O_n$ and $U_n$
I'm interested in characterizing conjugacy classes inside $O_n$ and $U_n$ over local fields of positive characteristic ($\neq 2$). I need this for my research on representation theory of these groups.
...
- 11
1
vote
0
answers
37
views
Attractivity of a system with state-dependent transitions
Let $A\in\mathbb{R}^{n\times n}$ and consider the following dynamical system:
$$
\frac{\mathrm{d}x(t)}{\mathrm{d}t} = -x(t)+\max\{0,Ax(t)\}, \ \ \ \ x(0)\in\mathbb{R}^n,
$$
where $\max\{\cdot\}$ acts ...
- 2,712
1
vote
0
answers
94
views
Generators for Ideals in ring of multivariate Laurent Polynomials
Consider the following problem:
Find an ideal $I \subset \mathbb{Q}[x^{\pm}_1,x^{\pm}_2,x^{\pm}_3]$ such that $I_{aff} \subset \mathbb{Q}[x_1, x_2, x_3] = I \cap k[x_1, x_2, x_3]$ requires more ...
- 11
1
vote
0
answers
90
views
When does a matrix with high rank have a minor with disjoint rows and columns and high rank?
This is a somewhat open-ended followup question to
Does an antisymmetric matrix with high rank have a minor with disjoint rows and columns and high rank? and Does a non-singular matrix have a large ...
- 20.2k
1
vote
0
answers
392
views
Pseudo-inverse of a column partitioned matrix
Given a $nm \times m$ matrix $A = \begin{pmatrix} A_1 \\ A_2 \\ \vdots \\ A_n\end{pmatrix}$ over $\mathbb{C}$, where $A_i$'s are $m \times m$ and $rank(A) = m$, is there an expression for the pseudo-...
- 11
1
vote
0
answers
120
views
Question about Local Henselian Rings
I have a question regarding properties/characterizations of local Henselian rings exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces":
Here the relevant excerpt:
Remark: ...
- 5,998
1
vote
0
answers
156
views
Does $AA^{-1}$ have the unique product property?
Let $A$ be a finite subset of a torsion free group $G$ with $|A|\geq3$. Does the set $AA^{-1}$ have the "unique product" property (i.e. there exist an element $c\in AA^{-1}$ that is uniquely written ...
- 2,118
1
vote
0
answers
152
views
Bound for Expectation of Singular Value
In my case, $X_{\boldsymbol{\delta}}\in\mathbb{R}^{d\times M}$ is a function of Rademacher variables $\boldsymbol{\delta}\in\{1,-1\}^M$ with $\delta_i$ independent uniform random variables taking
...
- 53
1
vote
0
answers
69
views
How exactly to adapt Brown's collapse from monoids to algebras?
In The Geometry of Rewriting Systems (Springerlink behind paywall), Kenneth Brown describes a method to collapse the bar resolution of a monoid. Roughly:
Given a simplicial set $X$ equipped with a ...
